Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2127 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C8 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#99 |
| commutator | C2×C4 | C2×C4×C8 |
| Frattini, cosocle | C22×M16 | C23 |
| socle | C22 | #1459 |
| prime radical | C22×#41 | C2×C4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C8 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C22×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
528 conjugacy classes of subgroups; only the 142 normal subgroups are shown.
This group has 528 conjugacy classes of subgroups — too many to draw. Only its 142 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2,4,3)(5,6,9,15,11,16,12,10)(7,17,19,13,21,28,20,29)(8,22,23,24,25,14,26,30)(18,32,33,36,35,31,34,27);(5,7,10,17,12,20,16,29,11,21,15,28,9,19,6,13)(8,18,24,31,26,34,22,36,25,35,30,32,23,33,14,27);(1,3,4,2)(5,8,9,23,11,25,12,26)(6,14,15,30,16,22,10,24)(7,18,19,33,21,35,20,34)(13,27,28,32,29,36,17,31)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=e,d^2=i,e^2=i,f^2=h,h^2=i,b^a=bd,c^a=ci,d^a=di,f^a=fgi,d^b=dg,g^b=gi,f^d=fi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72240000, #72240695, #72240698, … |
| maximal normal subgroup of | #72240000, #72240695, #72240698, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 35519);